Enumerating the Derangements of an $n$-Cube via M\"obius Inversion

In $\mathcal{L}$, the semilattice of faces of an $n$-cube, we count the number of automorphisms of $\mathcal{L}$ that fix a given subalgebra – either pointwise or as a subalgebra. By using M"obius inversion we get a formula for the number of derangements on the $n$-cube in terms of the M"obius function on the lattice of MR-subalgebras. We compute this M"obius function.